The problem of unit commitment (UC) involves determining configurations of generators to use for power generation so as to meet a target power demand. As used herein, a configuration includes Boolean variables that indicate whether generators should be OFF or ON for a particular time step. Generators typically include nuclear, thermal, and renewable power sources. Generators are subject to constraints such as stable operating level, rate of ramping up or down, and the amount of time the generator is ON or OFF, which makes the commitment (UC) problem a difficult combinatorial optimization task, which arises when the operation of N individual generators is configurated over T time steps, such that the total cost of producing electrical energy that meets the target power demand is minimized, while simultaneously observing the operational constraints of individual generators.
Conventionally, the unit commitment problem is typically formulated as a deterministic optimization problem where the outputs of the generators are assumed to be fully dispatchable, e.g., fossil-burned, nuclear, and the future power demand is assumed to be completely known or predictable. Various combinatorial optimization methods are known for solving deterministic UC problems, including methods based on dynamic programming, Lagrange relaxation, and mixed integer programming.
However, those assumptions are hardly true. Future power demand can rarely be predicted with errors less than 2% on prediction horizons of 24 hours or longer, so demand is in fact a random variable with at least that much standard deviation.
Moreover, the generation outputs of renewable undispatchable power sources, such as wind and solar, are highly volatile. For instance, the electricity generated by a wind turbine varies strongly with the wind speed, in combination with many factors such as its rated maximum power, cut-in and cut-out speed, generator efficiency, and air density.
Given these factors, it is more realistic to assume that the generated output is a random variable instead of a fixed value. Several different methods deal with uncertainties in the power demand and outputs of the undispatchable generators. One method plans for a higher than expected demand, in hope that the safety margin in power output would be able to cover possible deviations from target demand. The safety margin can be determined from the statistical properties of the demand, if available. However, that results in operating more and/or larger generators than are necessary to meet the target demand. That method essentially solves a non-deterministic problem via a deterministic approach in a conservative manner in hope that the overcommitted capacities can accommodate most, if not all, possible demand and generation output realizations.
Alternatively, another method handles uncertainties in demand directly and solves the corresponding non-deterministic decision problems via stochastic optimization methods, see Takriti et al., “A stochastic model of the unit commitment problem,” IEEE Transactions on Power Systems, 11(3), 1497-1508, 1996. By modeling and planning for all possible contingencies, a stochastic scheduler correctly handles future variations of supply and demand, and provides a safety margin implicitly. However, the model for representing stochasticity is limited to only a few scenarios.
Another method organizes the scenarios through an efficient probabilistic representation in the form of a factored Markov decision process (fMDP) that can naturally model the evolution of power demand and uncertain outputs of non-dispatchable generators, see U.S. Ser. No. 12/870,703, “Method for Scheduling the Operation of Power Generators,” filed by Nikovski et al., on Aug. 27, 2010. An approximately optimal policy for the resulting fMDP can be determined by a decision-space approximate dynamic programming (DSADP) method that achieves a better trade-off between the costs and operating risk than the deterministic approaches. However, that DSADP method uses AND/OR-trees that grow exponentially in the decision horizon, typically between 24 and 168 time steps, each of duration one hour, thereby rendering it impractical for most UC applications. In addition, that method uses a decommitment solver to select candidate configurations.